Linear Combination: A linear combination of a set of vectors S={v1​,v2​,…,vn​} in a vector space V is a vector of the form c1​v1​+c2​v2​+⋯+cn​vn​, where c1​,c2​,…,cn​ are scalars.
Span: Let S={v1​,v2​,…,vn​} be a set of vectors in a vector space V. The set of all linear combinations of the vectors in S is called the span of S and is denoted by Span(S).
Linear Independence: A set of vectors S={v1​,v2​,…,vn​} in a vector space V is called linearly independent iff the only linear combination of the vectors in S that equals the zero vector is the trivial linear combination, that is, c1​v1​+c2​v2​+⋯+cn​vn​=0 implies c1​=c2​=⋯=cn​=0.
Zero vector cannot be an element of an independent set.
Example: X={[01​],[01​]}
X is linearly independent set since c1​[01​]+c2​[01​]=[00​] implies c1​=c2​=0, ∀c1​,c2​∈R(F).
Span(X)=R2
Example: Consider the linear space of polynomials with degree n≤2. Let subset S={p1​,p2​,p3​} where p1​(t)=1, p2​(t)=t, p3​(t)=t2. Is S linearly independent?  Proof: Let a1​,a2​,a3​∈R,
a1​p1​(t)+a2​p2​(t)+a3​p3​(t)=0
a1​+a2​t+a3​t2=0
a1​=a2​=a3​=0
Hence S is linearly independent.
Example: S={cos(t),sin(t),cos(t−π/3)}  Proof: Let a1​,a2​,a3​∈R,